WORKS OF PROF. H. W. SPANGLER PUBLISHED W JOHN WILEY & SONS. Valve-Gears Designed as a Text-book giving those parts of the Theory of Valve-gears necessary to a clear under- standing of the subject. 8vo, xii + 17^ pages, 109 figures, cloth, $2.50. Notes on Thermodynamics. The Derivation of the Fundamental Principles of Thermodynamics and their Application to Numer- ical Problems. Fourth Edition, izmo, vii+7S pages, 24 figures, cloth, $1.00. Elements of Steam Engineering. By H. W. SPANGLFR, Whitney Professor of Dynam- ical Engineering in the University of Pennsylvania, ARTHUR M. GREENE, Jr., Professor of Mechanical Engineering in the University of Missouri, and S. M. MARSHALL, B.S. in E.E. Second Edition, Revised and Enlarged. 8vo, v + 297 pages, 290 figures, cloth, $3.00. VALVE-GEARS BY H. W. SPANGLER, Whitney Professor of Mechanical Engineering in the University of Pennsylvania. te fcg tje ^euiuc Utagram. ONE HUNDRED AND NINE ILLUSTRATIONS, SECOND EDJTION, REVISED AND ENLARGED. FOURTH THOUSAND. NEW YORK : JOHN WILEY & SONS, LONDON: CHAPMAN & HALL, LTD. 1 92 <3 \~ ; \; COPYRIGHT, 1890, BY H. W. SPANGLER. hp Srirnttfir flrce* Drummonft anb (Company PREFACE. THE writer, needing a book for class use which would give in one volume those parts of the theory of valve-gears necessary to a clear understanding of the subject, has pre- pared the following work. All the standard text-books on the subject, the current periodicals, and working drawings have been called on for data and methods, and the works of Zeuner, Auchincloss, Rankine, Whitham, Halsey, Marks, Reuleaux, Bilgram, and the files of Engineering and the Engineer have been freely used in preparing the text ; but the matter has been put in its present shape by the author. A few of the methods are original, but others confronted with the same problems have probably solved them in the same or in a better way. The designing of valve-gears is entirely a drawing-board process ; and in all but radial gears, and to a great extent even there, the actual method of laying down the work is given. The mathematical proof of the methods and results used is given whenever possible. The problems are in most cases made up from the data of engines actually in use. H. W. SPANGLER. UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA, PA., August 20, 1890. iii CONTENTS. CHAPTER I. PLAIN SLIDE VALVES. 1. Plain sliue-valves, .......... r 2. Method of action of valve, ......... 2 3. The eccentric, 2 4. Valve seat, face, and ports, ......... 3 5- Lap, 4 6. To determine position of valve and piston, ..... 4 7. Distance valve has moved from its central position, .... 5 8. Yoke-connection, 5 9. Vaive-diagrams, 6 10. Angle between crank and eccentric 90, 7 CHAPTER II. THE ZEUNER DIAGRAM. 11. To draw the valve -diagram, 10 12. Point of admission, .......... 10 13. Angular advance, n 14. Lead, ............. 12 15. From a given engine to draw the diagram, . . , . . 13 1 6. Distribution of steam as shown from the diagram, .... 14 17. Separate diagrams for each end of the cylinder, ..... 14 CHAPTER III. OVERTRAVEL AND PROBLEMS. 18. Overtravel, 19 19. Problem i, given r, d, cut-off and exhaust closure, .... 19 3. Problem 2, given lap, exhaust lap, lead and cut-off, . . . .20 v VI CONTENTS. 21. Problem 3, given cut-off, angle of lead, port and overtravel, , 21 22. Problem 4, given cut-off, lead and port-opening, . v . 22 23. When the piston and eccentric rods do not travel on parallel lines, . 23 24. To determine the position of the eccentric, . . , . 24 25. Effect of changing dimensions, . . . . , . 25 CHAPTER IV. MODIFICATIONS OF THE PLAIN SLIDE-VALVE. 26. Double-ported valves, . . . . . . . . .28 27. Allen or Trick valve, 28 28. Piston-valves, 30 29. Taking steam inside, 30 30. Two or more valves, 31 CHAPTER V. EQUALIZING CUT-OFF, LEAD, COMPRESSION, AND RELEASE. 31. Equalizing cut-off, . 34. 32. Equalizing cut-off and lead, . . . . . . % . 35 33. Equalizing exhaust and compression, 36 34. Circular diagram for determining movement of piston, . . . 37 CHAPTER VI. DESIGNING AND SETTING VALVES. 35. Designing a plain slide-valve, 40 36. To determine approximate solution, ....... 41 37. Equalizing lever, ... ...... 43 38. To put the engine on the centre, ....... 45 39. To set the valve 45 CHAPTER VII. THE STEPHENSON LINK. 40. The link, 47 41. Point of suspension, 48 42. Slip of block 48 43. Radius of the link, .... .... 50 44. Kinds of links, 50 CONTENTS. Vit CHAPTER VIII. THE VALVE-DIAGRAM. 45. Travel of the valve, . 52 46. The valve-diagram, 55 47. Curve of centres, c6 48. To lay down the valve-diagram, . . . . . . . .56 49. The virtual eccentric, ^g 50. Designing the gear, 58 51. Valve-stem and eccentric-rod, 58 52. Length of link, .50 53. The hanger, 59 , 54. Link suspended at bottom or centre of chord, 60 55. Open and crossed rods, . .61 CHAPTER IX. EQUALIZING LEAD AND CUT-OFF. 56. Equalizing lead, , . 63: 57. Equalizing cut-off, .......... 65 58. To lay down the motion, ......... 67 59. To lay down the centre of the travel of the valve, .... 67 60. To determine the centre of suspension of the hanger, .... 68 61. Position of stud, . 68 62. Reducing slip, . 70 63. Error of the Zeuner diagram, 70 CHAPTER X. THE GOOCH MOTION. 64. The Gooch link, . . 75, 65. Movement of the valve, 76 66. Constant lead, 78 67. Radius of link, 78 68. Suspension-rod, 79 69. The hanger, ........... 80 70. The valve-diagram, 80 71. To design a Gooch motion, ....* 82 CHAPTER XI. THE ALLEN AND FINK MOTIONS. 72. The Allen link-motion, 85 73. The valve-diagram, 8s Vlll CONTENTS. 74. The Fink motion, . . . ...... . 87 75. Radius of link, . . . * . . . . . . . . 87 76. Suspension of link, . . . . . .- . . , . 89 77. Movement of the valve, . . . . . . . . 89 78. The valve-diagram, . . . . . . ... 91 79. Radrus-rod at a fixed point in the link, 92 80. Hanger for radius-rod, ......... 92 Si. Setting the eccentric, . . . . . . . . 93 82. Designing, ............ 94 33. The Porter- Allen motion, .". 94 CHAPTER XII. SHAFT REGULATION. 4. Throttling governors, 98 55. Changing angular advance, ......... 98 56. Changing the eccentricity, ......... 99 87. Changing eccentricity and angular advance, 99 88. Erie governor, ........... 100 Sg. Armington and Sims, . . 101 90. Ball, 104 <)i. The valve, ............ 105 CHAPTER XIII. RADIAL GEARS HACKWORTH'S. 92. Radial gears, ..... ..... 107 93. Hackworth's gear, .......... 107 94. Constant lead, ........... 108 95. Movement of the valve, 108 96. The valve -diagram, 109 97. To design the gear, no 98. Right-hand rotation, . ........in 99. Errors of the diagram, . in TOO. Port-opening, . .......... 113 lor. Connecting up a Hackworth gear, 113 102. Attaching valve-stem outside, . . . . . . . .113 103. Equalizing port-opening, 113 104. Equalizing the cut-off, . . . . . . . . .115 CHAPTER XIV. RADIAL GEARS MARSHALL, ANGSTROM, AND JOY. 05. Marshall's gear, .......... 117 106. Errors of the Zeuner diagram, . . . .. . . .117 CONTENTS. IX 107. Proportions of the gear, . . . . . . . . .119 108. Designing, ........... 119 109. Angstrom's gear, . . . . . . . . . .120 no. The diagrams, ........... 121 in. Advantage and disadvantage of radial gears. ..... 121 112. The Joy gear, 121 113. Movement of the valve, ......... 123 114. Errors of the Zeuner diagram, ........ 124 CHAPTER XV. DOUBLE VALVES GRIDIRON VALVE. 115. Kinds of double valves, ......... 127 116. Gridiron valve, ........... 127 117. Polonceau valve, .......... 128 118. Diagram for gridiron valve, ........ 129 119. Combined diagrams for both eccentrics, . . . . . .130 120. Limits of cut-off, .......... 131 121. Width of ports, ........... 132 122. Angle of advance, .......... 132 123. Varying cut-off, . . 133 124. Arrangement used for varying cut-off, ...... 134 125. Width of cut-off valve, ......... 134 126. Varying width of block, . . .135 CHAPTER XVI. RELATIVE MOVEMENT POLONCEAU GEAR. 127. One valve on the back of another, . . . . . . . 137 128 Relative valve-circle, .......... 138 129. To draw the relative valve-circle, ....... 138 130. The Polonceau valve, ......... 139 131. The Polonceau gear, .......... 139 132. Valve-diagram, ........... 139 133. Limits of cut-off, .......... 140 134. Dimensions of valve, ......... 141 CHAPTER XVII. BUCKEYE GEAR. 135. The valve, 143 136. The eccentrics and connections, . . . . . . . 143 137. Movement of the valves, ......... 144 X CONTENTS. 138. Cut-off valve-diagram, .... 3"'. f- . . . 145 139. Changing cut-off, " , . -. _ ;^ . 145 140. The governor, . . . . . . .*. 146 CHAPTER XVIII. MEYER VALVE AND GUINOTTE GEAR. 141. The Meyer valve, . . . ... . . . * .148 142. Changing the distance between the blocks, ..... 148 143. Designing a Meyer valve, 150 144. Length of cut-off blocks .151 145. Cut-off with inside edges, . . . . . . . . .151 146. Guinotte's gear, . . . . . . . .''*- 152 147. Movement of the valve, . . . . . . . 153 148. To draw the valve diagram, . .154 CHAPTER XIX. BILGRAM, REULEAUX, AND ELLIPTICAL DIAGRAMS. 149. Bilgram diagram, 157 150. Problems, . . . . . . . . . 158 151. Reuleaux's diagram, . 160- 152. Problems " . . 161 153. Elliptical diagrams, 165 154. Velocity of the valve, 164 CHAPTER XX. CORLISS VALVE-GEAR. 155. Hamilton-Corliss engine, 166 156. Movement of the valve, . . 168- 357. Proportioning the parts, ....... . 169 SYMBOLS. A. Aoscissa of the end of that diameter of the valve-diagram which passes through the origin. B. Ordmate of the end of that diameter of the valve-diagram which passes through the origin. L. Distance from outside to outside of ports in top of Meyer valve. ./?. Radius of crank. a. Length of eccentric-rod in Fink gear to attachment of sus- pension-rod. b. Length of connecting-rod. Length of eccentric-rod in Fink gear from attachment of suspension-rod to link. .. c. One half the chord of the link. Distance from cross-head to point in connecting-rod at which radius-rod in Joy gear is attached. t. Width of port in upper valve-seat of double valves, y. Width of port in upper valve of double valves. g. Length of eccentric-rods. g v Length of radius-rod in link-motions. g- 2 . Length of valve-stem. h. Length of hanger in link-motions. /. Lead. /. Lap. /j. Length of radius-rod from closed curve to open one in radial gears. / a . Distance from end of valve connecting-rod to open curve in radial gears. / 3 . Distance from point on connecting-rod to point on open curve in the Joy gear. xi xii SYMBOLS. / 4 . Distance from point on connecting-rod to point of attach- ment of secondary radius-rod in Joy gear. n. Revolutions of the engine per minute. /. Port-opening. r. Radius of eccentric or eccentricity. fj. Throw of eccentric moving second or cut-off valve; and in link-motions, where unequal eccentricities are used, of the second one. r x . Diameter of the relative valve-circle with double valves. s. Distance the cut-off valve has to move from its central posi- tion to close the port. u. Distance between the centre of the link and the end of the valve-stem or radius-rod in Stephenson, Gooch, and Fink links/ In Allen motion distance end of radius-rod has moved from its central position. ,. In Allen motion, distance the link has moved from its cen- tral position. x. Distance the valve has moved from its central position for any position of the crank. y. Half the distance between the blocks in a Meyer gear. of. Fixed angle between the crank and the eccentric, angle be- tween eccentric-rod and the centre line of its motion in a Fink motion, and angle of path of end of radius-rod in radial gears. 0. Ninety degrees less than the angle between the dead-point and point of cut-off. y. | Angles used in finding movement of the valve in link-mo- y'. \ tions. 6. Angular advance of main eccentric. ) = r s i n (<* +<) (4) Evidently, if GO = 90 6, x = r. In Fig. 9 the same diagram has been drawn as in Fig. 8 r but the diameter of the valve-circle has been moved through an angle bad = <5. O In this figure, if ac = x = r sin (tf+ GO), the circle still represents the valve-diagram. For ad = r and dca is a right angle; .*. ac ad cos doc = r cos (90 d GO) r sin ($ -{- GO) = x. It is to be remembered that the valve-diagram shows the movement of the valve for varying positions of the crank, and the centre line of the valve-diagram as ad in Fig. 9 is not the position of the eccentric when the crank is at an. The small diagram shows the relative position of the crank and eccentric to give the valve-diagram shown in the figure. 12 VALVE-GEARS. We see that by giving the eccentric angular advance we have simply moved the valve-circle about a through the angle 6. It will be seen that the port now opens when the FIG. 9. crank is at ak, and therefore steam-pressure is on the steam- piston at the beginning of the stroke. The angle hak is often called the angle of lead. 14. Lead. When the crank is on the dead-point an, the distance hn is evidently the opening of the port. This dis- tance is called the lead. The lead can therefore be defined as the amount the oort is open to steam at the beginning ot 8 *--4 FIG. 10. the stroke, or when the crank is on its dead-point. In the same way exhaust-lead is the amount the steam-port is open to exhaust at the beginning of the stroke. THE ZEUNER DIAGRAM. 15. From a given Engine to draw the Diagram. Hav- ing any valve given moved by a single eccentric, we can lay down the valve-diagram and determine the points of admission and cut-off, the opening and closing of the ex- haust, etc. In Fig. 10, suppose the valve to be a inches over all and b inches inside, the steam-port to be c inches wide, the ex- haust-port d inches and the bridges or material between the ports to be e, and the eccentricity to be r inches, and the angular advance d degrees, to determine the various points relating to the valve. From the figure the lap = 2e + d - b a d 2e 2C and the ex. haust-lap = . To draw the diagram in Fig. n 14 VALVE-GEARS. draw the two lines ab and cd at right angles to each other. Let od be one dead-point, the engine to turn in the direction of the arrow. Lay off the line oe so that the angle aoe is d degrees. On oe lay off of= r the eccentricity, and on of as a diameter draw the valve-circle. With o as a centre and og as a radius equal to the lap, draw the lap-circle gqk, and with a radius oh equal to the exhaust-lap draw the exhaust- lap circle /is I. 16. Distribution of Steam as shown from the Diagram. When the crank is at od, the steam-port on, say, the right side is open the distance mp or the lead, while the exhaust- port on the opposite side, say left, is open np or the exhaust- lead. When the crank reaches oe both ports are opened widest, that on the right to steam and on the left to exhaust. When the crank reaches ok the steam-port on .the right closes and cut-off takes place. When the crank reaches ol the exhaust closes on the left side of the piston and com- pression begins. When the crank reaches oh' (oh prolonged) the exhaust opens on the right side of the piston. At og' (og prolonged) the steam-port opens on the left. At oc the steam-port is open on the left a distance/?//, and the exhaust is open on the right a distance pn. When the crank reaches ok' the steam is cut off on the left-hand side, at ol' the exhaust closes on the right-hand side. At oh the ex- haust on the left opens again, and at og the steam is again admitted on the right-hand side. 17. Separate Diagram for each End of the Cylinder. To make this clearer, the diagrams Figs. 12 and 13 are drawn, one of which shows the distribution of steam in the left-hand end of the cylinder, while the other shows the dis- tribution in the right-hand end. The letters refer to the same things as in Fig. 1 1. The circles with oa as a radius are drawn to any convenient scale to represent the line travelled through by the crank-pin, and the lines ab represent the stroke of the piston to the same scale, gk and g'k' are the lap-circles, hi and h'l are the exhaust-lap circles. In the right-hand end of the cylinder, Fig. 12, starting at THF ZEUNER DIAGRAM. the dead-point a, the steam-lead is mp, cut-off takes place at 2, exhaust opens at 3, and the port is open the distance rip' or the exhaust-lead at the beginning of the return stroke ; at 4 the exhaust closes, and at i the steam-port opens again so FIG. 12. that when the crank again gets to a the port is open the lead. On the lower line ab, representing the stroke of the piston, starting at a, steam is admitted until the piston reaches 6, when cut-off takes place ; at 7 the exhaust opens, and the piston travels to the end of the stroke. On the return stroke, at 8 the exhaust-port is closed, at 5 the steam- port again opens, and the piston travels to the end of its stroke again. Steam is being admitted while the crank 16 VALVE-GEARS. travels from i to 2, it is being- expanded from 2 to 3; exhaust is taking place from 3 to 4, and compression from 4 to i. While this is taking place in the right-hand end of the cylinder, the left-hand end is also receiving steam and ex- hausting, as shown in Fig. 13. Starting at the same dead- point a, the ex-haust is open on the left a distance np equal FIG. 13. to the exhaust-lead ; at 12 the exhaust closes, at 9 steam opens on the left, and at the dead-point b the port is open to steam a distance m p' equal to the steam-lead. On the return stroke, at 10 steam is cut off, at 1 1 exhaust opens, and at a the dead-point is reached. The line ab at the bottom of the figure shows the move- ment of the piston. Moving from a, at 16 exhaust closes, at THE ZEUNER DIAGRAM. I/ 13 steam opens; on the return stroke, at 14 cut-off takes place, and at 15 exhaust opens. Exhaust takes place while the crank is moving from 11 to 12, compression from 12 to 9, admission from 9 to 10, and expansion from 10 to n. QUESTIONS. 14. Explain the method of drawing the Zeuner diagram. 15. How is the point of admission found? 1 6. What is the port-opening for any position of the crank ? 17. What is angular advance, and why is it given? 1 8. What effect has angular advance on the valve-dia- gram ? 19. What is lead? 20. What is meant by angle of lead ? 21. What dimensions are required to determine the valve- diagram for a given engine? 22. Explain fully the different events occurring in one end of the cylinder, in the order of their occurrence, during one complete revolution. 23. Explain fully the events occurring in both ends of the cylinder, in the order of their occurrence, during one complete revolution. 24. How determine the actual position of the piston, the length of the connecting-rod being given, for each event occurring in one revolution in one end of the cylinder ? PROBLEMS. 4. Given the eccentricity 3 inches and the angle between the crank and the eccentric 90, how far has the valve moved from its central position when 09=30, 60, 90? Draw Zeuner diagram and scale distances. 5. In the above problem, if the lap is if inches, through what angle has the crank moved from its dead-point when the port opens? What when the port closes? 6. What is the port-opening when & = 75, and what when GO = 150? 1 8 VALVE-GEARS. 7. A given engine is 136 inches from centre of shaft to centre of exhaust-port. The valve is Q| inches over all, and 5fJ- inside. The exhaust-port is 3 inches, the steam-ports each if inches, and the bridges if inches each, r = 2j". The length of the connecting-rod is 90 inches, and the eccentric-rod 60 inches. The cylinder is 18 inches diameter by 24 inches stroke, and the angle of advance is 25. Where in each stroke is steam admitted and cut off, and where does exhaust and compression in each end take place? Measure all dis- tances for each end of the cylinder from that end of the stroke at which steam is admitted. CHAPTER III. OVERTRAVEL AND PROBLEMS. 18. Overtravel. The travel of the valve is otten more than sufficient to open the port wide. The excess is called overtravel. Thus in Fig. 11, the distance qr is the width of the port, and the port is wide open to steam while the crank passes from ox to oy, and the distance rf is the over- travel. Similarly st is the width of the port, and the port is wide open to exhaust from ov to ow. That is, on the right side the port begins to open at og, is wide open at ox, begins to close at oy, and is entirely closed at ok. In the same way the exhaust on the left-hand side begins to open at oh, is wide open at ov, begins to close at ow, and is entirely closed at ol. Without overtravel the valve was opening the port from og to oe\ with overtravel the port is opening while the crank passes from og to ox. Over- travel therefore causes the port to be opened and closed more quickly, but for the same opening of the port the eccentricity must be greater by the amount of overtravel. The following are the ordinary problems that occur in designing valves. 19. Problem I. Given the eccentricity r, the angle of advance d, the point of cut-off, and the point of closing of the exhaust, to find the lap, exhaust-lap, lead and exhaust- lead, and the greatest possible opening of the port. In Fig. 14 draw ao and do at right angles to each other. Lay off aoe = $, the angular advance. On oe, with a radius equal to one half the eccentricity ( J, draw the valve-circle. 20 VALVE-GEARS. Draw ok to represent the position of the crank at cut-off, and ol for the closing of the exhaust. Through the inter- section of these lines with the valve-circle, and with o as a centre, draw Ish and kqg. Then ol is the exhaust-lap, ok the FIG. 14. lap, mp the lead, np the exhaust-lead, qf is the greatest possible opening of tne port to steam, and sfto exhaust. 20. Problem 2. Given the lap, exhaust-lap, point of cut-off, and the steam-lead, to determine the eccentricity and angle of advance. In Fig. 15 draw 0r more than two values of u on each side of u = o? 92. Why is the stud often placed back of the arc of the link? 93. How can the point be determined? 94. Is the slip greater or less than if placed on the arc? 95. Is there any advantage in so placing the stud? 96. How determine the length of hanger which will reduce the slip ? 97. In the Zeuner diagram what are the errors, and is admission, cut-off, or maximum port-opening most affected thereby? 98. Do the errors of the diagram tend to neutralize or to increase the variation in cut-off due to the angularity o* the connecting-rod? EQUALIZING LEAD AND CUT-OFF. 73 PROBLEMS. 32. Given in a Stephenson link with open rods r = Si inches, eccentric-rod 75^ inches, 2c 26 inches ; steam-lap 2$ inches, cut-off for u c at .72 stroke. What is tf, and to what angles should d be changed for constant lead at u = o and u = c? 33. What should be the radius of the link \{2c= 12 inches, r = 2\ inches, length of eccentric-rod = 56 inches, and the angle of advance 3 = 16, if the lead at full and mid gear is to be equalized. 34. In a Stephenson link-motion with the eccentric-rod attached back of the link having the following data determine the position of the link for u = 4 inches, GO = 40 : Eccentricity 2j inches, angular advance 18, eccentric-rod 56 inches to the arc and 53^ inches to the point of attachment to the link: radius of the link 55 inches, 20 = \\\\" ; centre of tumbling- shaft 39" from centre of shaft and 1 1 inches above centre of the engine ; length of hanger 14^ inches ; arm of tumbling- shaft curved so that when the link is in its middle position the upper end of the hanger is 9^ inches above the centre- line of the engine arid 56 inches from the shaft ; the point of suspension of the link being T 9 inch back of the centre of the arc, stroke 24 inches, and connecting-rod 89^ inches. 35. With the above data find the centre of the travel of the end of the valve-stem. 36. With the above data as to the HIIK, and with the length of hanger 14^- inches, find the position of the tum- bling-shaft and length of the arm so that the cut-off is equal for both ends at .5 and .8 stroke. 37. Taking the link data as above, find the point of cut- off in one end corresponding to -J stroke in the othei end, and after changing the data so that the radius of the link is 56 inches, the centre of suspension is at the centre of the arc of the link, and the upper end of the hanger is at the same point as before, the other data remaining the same, examine the cut-off for the same position of the tumbling-shaft arm 74 VALVE-GEARS. and determine whether the changing of the radius of the link and point of suspension in this case has changed the cut-off. 38. In the problem thus given is 14^ inches the best length of the hanger that could have been taken ? If not, determine the best length to reduce the slip as much as possible for full gear, the link to be supported -f$ inch back of centre of arc. 39. Draw a Zeuner diagram for Problem 34, and lay out the actual movement of the valve for u c, u = , and u = o. 40. Replace the single eccentric in Problem 27 by a link- motion which, when in full gear, will fulfil the same condi- tions. The length of the hanger is not to be greater than 2c, and cut-off is to be equal at .5 and at .8 stroke for the two ends of the cylinder. CHAPTER X. THE GOOCH MOTION. 64. The Gooch Link. Fig. 45 represents a centre-line diagram of a Gooch link-motion, ab is the crank ; ac and ad are the eccentrics set at equal angles of advance ; ce and df arc the eccentric-rods; e/is the link which is curved in FIG. 45. the opposite direction to the Stephenson link ; kl\s a suspen- sion-rod which supports the link at its central point and is attached to a fixed point / on the engine-frame ; gj is the radius-rod, the end g sliding in the slot of the link, and the end/ being attached to the valve-stem. The end ^is moved up and down in the link by means of the hanger km, the lower end m of which is attached to the radius-rod and the upper end h is attached to one arm hi of a bell-crank pivoted at i. When g and e are together, the eccentric c drives the valve, and the engine turns in the direction of the arrow. When the radius-rod is lowered so that /and g are together, 75 76 VAL VE-GEARS. the eccentric ad drives the valve, and the engine turns in the opposite direction. 65. Movement of the Valve. The method followed in deducing the equation for the movement of the valve is identical with that followed for the Stephenson link-motion. In Fig. 46 the crank and gear are represented when the crank has moved an angle GO from its dead-point and the radius-rod has been raised a distance u. The distance the point e has moved from its central posi- tion along the line ae is an r cos (90 GO d y) r sin (GO + d -f- y). The distance the point g has moved corresponding thereto is r (sin (co-\-d 4- y)) cos y. In the same way the point f has moved from its central position ap = r cos (90 d y -f- GO) = r sin (d -f- y GO) and g has moved a corresponding distance -- r sin (tf + y a?) cos r, and the motion of g from its central position is therefore x = r sin (GO -\- d -(- y) cos y -| r sin (# -f- 7 G?) cos ;/. As in Fig. 45 the radius- rod gj is long as compared to gk or ke, the horizontal motion of j is practically the same as of g, and we have the above value of x for the movement of the valve from its central position. THE GOOCH MOTION. 77 7$ VAL VE- GEARS. Expanding the second member of this equation, and c ing cos y = i, sin y = -, we have x = r f - cos ^ -f- sin d\ cos GO -| I cos 6 sin tf) sin o>. This equation can also be put in the same form as for a sim- ple valve. If we let r I sin tf + - cos tfj = A o and (costf--sind)=, we have x = A cos G? -|- ^ sin 00, which is the same equation as for a simple valve. 66. Constant Lead. An examination of the equation for x will show that the value of A is constant whatever the value of u\ that is, for every value of u the valve-circle crosses the line representing the dead-point at the same point, or the lead is constant. 67. Radius of Link. When the engine is on the dead- point, as shown in Fig. 45, if the lead is constant, the point j must not change position as the radius-rod is raised or lowered. The link must therefore be drawn with/ as a centre and jg as a radius, or the radius of the link is the length of the radius-rod. The eccentric-rod and the radius- rod should each be as long as possible, but a better distribu- tion of steam is generally obtained by making the eccentric- rod the longer. The same statements as to the length of the link apply here as in the case of Stephenson's link. The distance from the centre of the shaft to the centre of the exhaust-port can be approximately determined as follows : The mean position of the chord of the link is approximately THE GOOCH MOTION. 79 at a distance g from the centre of the shaft. The distance from the chord to the arc is , approximately. The dis- O 1 tance from the centre of the shaft to the centre of the arc of the link when the arc is in its middle position is*?- -. Adding to this the length of the radius-rod g^ and the length of the valve-stem to the middle of the valve r s , we have - -\-g l = the distance from the centre of the shaft to the centre of the exhaust-port. This is, of course, only an approximation, but it is correct enough for practical purposes. 68. Suspension-rod. The point of support of the sus- pension-rod for the link should be in such a position that the rod swings equally on each side of a vertical line. Fig. FIG. 47. 47 gives the two positions of the link when the crank is on the dead-points. bd be -f- cd r sin d + g, approximately, and be = ib -\-te= r sin d -\- g, and the mean position is at /=: from the centre of the shaft. As the hanger is attached to the arc, and not the chord, the distance 80 VAL VE-GEARS. If then the suspension-rod is attached to the link at the centre of its arc and to a fixed point whose distance from b f \s g parallel to ad, and the length of the suspension- <5 i rod above ad, the attachment will give the proper motion. The distance above ad is usually given as the length of the suspension-rod, but this brings the arc in which the centre of the link swings entirely above ad, while to keep the mo- tion as nearly correct as possible the line ad should inter- sect the arc, and the point g of the link should swing equally above and below this line. 69. The Hanger. To determine the arc in which the point h of Fig. 45 should move. From the point // in Fig. 47, just determined, lay off the length of the radius- rod to the right in the figure. Mark on this distance the point m of Fig. 45. Through the point thus determined lay off a distance mh at right angles to ad. Through this point draw an arc whose centre is on the right side of the link whose radius is the length of the radius-rod, and whose centre lies at a distance equal to the length of the hanger above aj, Fig. 45. This arc should lie equally above and below the line through the centre of the arc, parallel to aj\ and is the curve in which h should move. Practicably this is much too large, and this arc would be replaced by one of much smaller radius, the centre for which would be deter- mined in the same way as shown for a Stephenson's link in Fig. 36- 70. The Valve-diagram. If all the data relating to a Gooch motion is given, the valve-diagram can be laid down as follows : Suppose the lap, angle of advance, throw of the eccentric, length of link, eccentric-rod, radius-rod, and dis- tance to the centre of the exhaust-port be given, to determine the point of cut-off, etc., for a given value of u. Draw ab and ac in Fig. 48 at right angles to each other, and lay off baf = d. Make ad = c, and ae = g, and eh = r. Draw the perpendicular ih to ae. Make the angle fak = dec, and lay off ak = ei. ak is then the diameter of the virtual THE GOOCH MOTION. 81 eccentric for u = c. Or, make af= r y fak = dea, and draw fk at right angles to af\ then ak is the diameter of the virtual eccentric. FIG. 48. The proof of the construction is as follows : For the co- ordinates of k we have, from page 78, A = aj = r \^sin d -|- cos d J, o B kjr (cos <5 - - sin d], cos ^ = tan akj =. r> - Q ~ C . C cos o sm d i tan S g As = tan tan # = tan d -f- tan or i tan d tan dea 1 akj = 6 -}- dea, sin akj = aj=r fsi sn ^ - cos g 82 VALVE-GEARS. = r sin 6 -\- r tan dea cos tf r sin 8 cos dea -f- r sin dea cos tf cos sin (dea -\ sin (dea r + #) cos dea or sin (dea 4- d) ? ak = r , r j 7-7 = - cos dea sin akj cos #> = r sec dfctf == ', as by construction. Draw the lap-circle qvs. The point of admission for u = c is then on aq, and of cut- off on as. The lead is rj. For any other value of u lay off JO" Ji tne point g so that -r -. Join and -, and draw the valve- 1 ft ^ circle with agas a diameter; au will now be the point of ad- mission, av the cut-off, and the lead rj will be the same as before. 71. To Design a Gooch Motion. We will suppose the same data to be given as in a Stephenson's link to design a Gooch motion, i.e., ports, point of cut-off, angle of lead, dis- tance from centre of shaft to centre of exhaust-port, to lay down the motion. The length of the valve-stem and link should first be determined or assumed. From the distance between the centre of the shaft and the exhaust-port should be taken the length of the valve-stem. The remainder is g-\-g, - - . If now g or g t is assumed, the other is determined. All these data must be fixed before trying to lay down the diagram. In Fig. 49 draw ab and acat right angles, and let as and aq be the positions of the crank for cut-off and admission. Bisect the angle saq by the line ak, and find the diameter ak of the virtual eccentric for the required port-opening, as already shown in Chapter III. Make ad = c and ac = g, and THE GOOCH MOTION. 83 lay off ei equal to ak. Draw ih parallel to ab. Then eh is the eccentricity. Make kaf = dea. Then the angle fab is the angular advance. We have now all the data required for laying down the motion, except the hanger for the link and radius-rod. The FIG. 49. only directions that can be given about them are to make them both as long as possible, and to support them, as pre- viously indicated. QUESTIONS. 99. Sketch a Gooch motion and describe it. 100. Deduce the equation for the movement of the valve, and prove that the Zeuner diagram represents it. 101. How does the lead vary with a Gooch motion? 102. Why is the radius of the link made equal to the length of the radius-rod ? 103. How determine the centre of suspension for the link? 104. What is the curve in which the upper end of the hanger must move ? 105. Explain the method of drawing the valve-diagram. 106. What data must be given in designing a Gooch motion ? Explain in full. 84 VAL VE-GEARS. PROBLEMS. 41. Draw the Zeuner diagram for the Gooch motion, having 8 = 20, r = 2f inches, 2c 12 inches, = 48 inches, lap = inch, stroke = 18 inches, connecting-rod 45". Deter- mine the point of cut-off for u = o, 2, 4, and 6 inches on both strokes. 42. If in the above problem the radius of the link is 36 inches, the hanger is attached 10 inches from the arc and is 1 8 inches long, and the arm of the tumbling-shaft is 24 inches, the centre being i/f inches below the centre-line of the engine and 82 inches from the shaft, draw the motion when oo = 40, u = 4 inches. 43. What must be the radius of the link, if the cut-off is to be exactly the same on both ends, for u 6 inches ? 44. Replace the link-motion of Problem 40 with a Gooch motion cutting off at .8 stroke on both ends, using such other data of Problems 40 and 27 as are required, but making the valve-stem not less than 20 inches long. 45. Taking the data given in Problems 42 and 43, show the errors of the Zeuner diagram for u = 4 and u 6 inches. CHAPTER XI. THE ALLEN AND FINK MOTIONS. 72. The Allen Link-motion is represented in Fig. 50, in which ab is the crank, ac and ad the eccentrics, de and /the eccentric-rods. The link fe is a straight one, supported at FIG. 50. its centre g by the hanger mg, which is attached to one end of a rocker-arm mlk turning around /. hj is the radius- rod supported by the hanger ki t the upper end of which is attached at k to the other end of the rocker mlk. By turn- ing mlk about its centre /, the link is lowered at the same time that the radius-rod is raised. 73. The Valve- diagram. The diagram for the valve- motion is the circle, as in the case of the other links that we 86 VAL VE-GEARS. have dealt with. The movement of the valve can be deter- mined in the same way, and is / . c* uu. \ = r ( sin d -(- - cos 6) cos GO . ur ( -{ -- ( c \ c(u u.} cos 6 -* - sin sin ug in which u is the distance the link has moved downwards from aj, while u l is the distance the end of the radius-rod has moved upwards. The hangers ki and mg should be of equal length, and as long as possible, mk should be equal to hi. The ratio of the parts ml and Ik is given by the equation and as Im -\- Ik =. ai, from the two equations the value of Ik and Im can be found. The position of / can be found by laying off along aj a distance and at right angles to^/'a distance equal to mgthe length of the hanger. To draw the valve-diagram for any value of #, the coordinates of the centre of the circle are A r ( . f-uu, \ == 5 (sin* + - costfj and B ur I c(u ,) . ur I - = 2 2C \ gU For any value of u, the corresponding value of u l is T I ./ ^ " ji X Im THE ALLEN AND FINK MOTIONS. 8/ The motion is but seldom used in this country, and it is unnecessary to go further into the details. 74. The Fink Motion. The Fink motion is represented in Fig. 51, in which R is the crank, od the eccentric directly opposite, the travel of the piston being supposed to be along ob^. dj\s the eccentric-rod, rigidly connected at/ to the link cd '. The point p' of the eccentric-rod is constrained to move in the line ob , or very nearly so, by the suspension- rod p'g attached to a fixed point on the engine-frame at g. mb f is the radius-rod, the end m sliding up and down in the link, while the end b' is attached to the end of the valve- stem b'b . The radius-rod is moved in the link by means of the suspension-rod et, the lower end e of which is moved in a suitable curve. If the crank is on a dead-point, the point j would be on ob Q , and nm would be vertical. 75. Radius of Link. As it is desired that the engine should have constant lead whatever the position of the radius-rod, if the valve does not change us position the radius of the link must be mb' , and this is what it is gener- ally made. An examination of the figure will show that, when the engine is on one dead-point, the distance from the centre of the shaft to the centre of the valve is and when the engine is on the other centre the distance from o to b is _ od+dj + mb' + b'b, = - r + (a + b) +& +^ 2 . That the lead may be equal for both ends of the cylinder with a common D valve, the distance from the centre of the shaft to the centre of the exhaust-port should be the mean of the two values of ob above found, or VALVE-GEARS. THE ALLEN AND FINK MOTIONS. 89 76. Suspension of Link. The coordinates of the point g can readily be determined. On one dead-point ov = od -\- dp' = r -f- a t and on the other dead-point ov' = od -\- dp = r -f- a, and *&gp r should swing equally on either side of pg, op should be the mean of these values of ov or a. To determine the pointy, lay off from p the distances/^ =. pv' = r. Draw an arc wj>' with the desired value of ' pg as a radius. Bisect the distance pv z at/\ (not shown) ; then the point p l should be on one point in the arc, and laying off p,g= the desired value of gp 1 gives g the centre of motion. By calculation, and Too great care cannot be taken in laying down this gear, as the approximation to the movement of the valve is much more crude in this case than in any that have preceded it. 77. Movement of the Valve. From the figure ob. = of+fp' + p'k + kl + IV + b'b. , of=r cos oo, fp' = a cos a, p'k = (b + x) cos <*, kl = y siri a, IV = Vg? - u\ b'b. - & , and ob = r cos co -f a cos a -f- (b + x) cos a -\- y sin a +St, '-'#+& 90 VALVE-GEARS. From the figure also y cos a = u -f- (b + x) sin or, or -f- (b + JT) sin . u sin a -|- ( -\- x) sin* and y , y sin a = J cos a -* b + u sin a -\- x (b -4- x} cos a -4- y sin a = - . COS a u * For Vg? -- i? we may put g^ , and making all the substitutions in ob^ , we have b 4- u sin a ob n = r cos oo 4- a cos a 4- - x u^ ' cos a: " ^ ' ^ 2 2^-, * As ^' is an arc of radius g^ , approximately y = 2gjc, r sin GO = a sin or. We have seen above that _u-\- (b -{- x) sin a COS Of and as ^ sin a is small, we can make u -4- b sin a y* (u -4- b sin a}* V and x = = COS Of 2g^ 2g l COS a or b + # sin a ob n = r cos GD -\- a cos # + COS a (u + 6*nar , , *1 2^-, COS 3 2^" We have above r sin G? = a sin # and sin a =. sin &?, cos I r* r 1 . i r a tf = A / i ^ sin 2 &? = i ^ sin a &?, = I + sin 2 ce?, y a 20* 'cos ' 2^ 2 t ALLEN AND FINK MOTIONS. 91 and - = i + , i sin 8 GO. ' 2 cos a and rur ubr\ oo ^ = r cos GO -f- f -|- j sin after collecting the terms. Now the movement of the valve for any value of GO is the distance the valve has moved from its central position. We have already seen that the central position is at ob, = a + + -,+-,; the movement of the valve is therefore the difference of the last two equations, or sin GO . ur( b\ x = r cos co -f- ^i -|- j si ~ a sin 3 GO. (A) The last term in this equation is generally small, and may be omitted ; and then ur = r cos GO -4- - ur i b\ ( l + J sin a?, which represents the movement of the valve for any value of u and GO. 78. The Valve-diagram. The coordinates of the centre of the valve-circle are and ( i 4- ); so that if we have 2 2a \ gj the data of the gear given, the point of cut-off, admission, lead, etc., can be determined from a diagram such as Fig. 52, which represents the valve-diagrams for four values of u. Evidently, if u = o, the diameter of the valve-circle is r, or VALVE-GEARS. the lap plus the lead equals the throw of the eccentric. An examination of the first value of x deduced (A), shows that the last term is less the greater the value of a and the FIG. 52. smaller the value of b. Generally, b is made very small, oftentimes zero, or the point/' in Fig. 51 is attached directly to j. When this is the case, = r cos . ur . -| --- sm GO. 79. Radius-rod at a Fixed Point in the Link. We have supposed in our deduction that u is kept constant, that is, that the end of the radius-rod moves in the link. If we had supposed that y was constant and that u changed, we would have found that the same equation would give the movement of the valve, putting/ in the place of u. 80. Hanger for Radius-rod. To determine the point of suspension of the hanger for the radius-rod. Referring to Fig. 51, and calling tb' =, we have But ts = - ; consequently I THE ALLEN AND FINK MOTIONS. 93 We have already seen that from o to the centre of the port is a -\- b -\- gi +" 2 ; consequently the position of s when the valve is at the middle of its travel is and as the suspension-rod should be vertical for this point, for the centre of its movement and he et st = h u, when h = length of suspension-rod. If we call y = oh and y^ oh = a -j- b -f- g l g Q for u o, and z = he and z 9 = he = h for u , then the eccentric has moved to the position oh and the point c to i and /to k. Draw the lines in, km, and hp at right angles to oc, and ir parallel to oc. Then km is the movement of the valve if the angularity of the valve-rod ks is neglected. Let be = ac hi =.l l ,fc= ki= /, . Call ico = a, km = x, and oh = r. Then x = km = kq -f- qm kq + in, RADIAL GEARS HACKWORTPTS. 109 But Now en = op r sin <*? approximately, and in = en tan a r sin GL> tan a, and r-# = hp in = r cos GO r sin 6? tan <* ; or substituting, we have kq = -f (r cos oo r sin GO tan <*) A and # = 7 (r cos oo r sin G^> tan <*) -f- ^ sin GJ tan a ; ^i or x = j r cos oo -[- sin &/ r tan a ~ r tan / - /. \ . 7 tan r I sm = A cos GL> -|- .5 sin 60, which is the equation to the Zeuner diagram in which A = l -f r and B = r^-^] tan a. 96. The Valve-diagram. As most engines to which FIG. 63. these gears are applied are vertical, the diagrams are drawn as though one dead-point was at the top of the figure and 1 1 VAL VE-GEARS. one at the bottom. In Fig. 63 is drawn the diagram for this motion, in which / 9 r r //, - /' oa = - and av = - - tan #. From these equations we see that whatever the value of ar, the centre of the valve-circle is on the line ab. Now ab /, - / 2 tan #0 = = -. tan a. ; oa / or as oba would be the equivalent of the angle of advance of a single eccentric, to produce the same distribution of steam tan d = 77 rf . (/, / 2 ) tan a For any value of #, therefore, the corresponding valve- circles can be drawn. With o as a centre and the lap od as a radius draw the lap-circle def. The port opens when the crank is at od, the lead is ec, and the cut-off takes place when the crank reaches of. 97. To Design the Gear. To design a Hackworth motion, suppose we have / : , / 2 , the lap, and the lead given, to determine the value of r and of a for a given cut-otf. In Fig. 64 draw oa and ch at right angles to each other. From o lay off oe lap and ef = lead. Draw oi to represent the position of the crank at the point of cut-off. Draw the lap- circle ehi. The valve-circle must pass through f, 0, and i. Finding the centre k, and drawing the circle, we have /, A X of of=jr or r = r - t-i *> t RADIAL GEARS HACK WORTH'S. Ill and /. / 2 / 2 tan fok tan fob = - L tan a or tan a = , 'a *i * which determine the other details of the gear. 98. Right-hand Rotation. For convenience in dealing with the errors of this diagram we will use the term " right- hand rotation" to mean as follows : Assume the end of the FIG. 64. vibrating link, Figs. 61 and 62, which moves in a closed curve to be at your left, the end which moves in an open curve or straight line to be on your right, and the valve to be above the vibrating-link. Rotation in the direction of the movement of the hands of a watch will be called right- handed. Then in Figs. 61 and 62 movement in the direction of the arrows is right-hand rotation. 99. Errors of the Diagram If a diagram is drawn according to the formulas deduced above, and a second dia- gram is made showing the actual position of the valve from a scale-drawing of the motion, it will be found that there is an error in the Zeuner diagram as applied to this gear. It will be found that the actual diagram approaches more nearly the theoretical for right-hand rotation, and if the diagram is laid down for different values of a, the smaller 112 VAL VE-GEARS. this angle the less the errors of the diagram, and in design- ing a motion of this kind these points should be remembered. FIG. 65A. FIG. 656. Figs. 65 and 66 represent the Zeuner and actual diagrams for a motion of the same proportions as Fig. 62 ; Fig. 65 A FIG. 66A. FIG. 66B. being for right-hand rotation and tan a = .75, and 656 being for right-hand rotation with tan a .4. Fig. 66 represents corresponding conditions with left-hand rotation. t RADIAL GEARS HACK WORTH'S. 11$ 100. Port-opening. While there is a difference in the amount of port-opening and cut-off, with motion in either direction, for the two ends of a cylinder, the difference is greater with left-hand rotation, and might become serious in a badly designed gear. 101. Connecting up a Hackworth Gear. In a horizon- tal engine advantage could be taken of this by making the more rapid valve-motion at that end of the cylinder at which, owing to the angularity of the connecting-rod, the more rapid motion of the piston itself occurs. In vertical engines designed to run left-handed it is well to introduce a rocker- arm, thus reversing the motion and giving the wider opening and retarded cut-off on the up-stroke. If desired, the eccen- tric can be changed 180, or the swinging link attached to the crank directly. The point i would then travel on the part ce instead of ci, Fig. 62, if the engine turned in the same direction, and a rocker would have to be used or steam taken inside the valve. 102. Attaching Valve-stem outside. Sometimes, in- stead of attaching the valve-rod at k between h and t t as in Fig. 62, it is attached at a point outside, as at k' . Our reasoning still holds good ; but the / now changes sign, and the equation to the movement of the valve is x = r cos GO -|- r( -) tan a sin GO. The sliding of the end i of the vibrating-link along the guide ie is liable to excessive friction as the angle a increases, and this led to the other types of simple radial gear. 103. Equalizing Port-opening. By the use of an equal- izing lever it is possible to make the port-opening on both ends of the cylinder the same for any given cut-off, and if desired the cut-off in the two ends of the cylinder can be made exactly equal. Fig. 67 represents the path of that point of the radius-rod 114 VAL VE-GEARS. FIG. 67. to which the valve-stem or valve connecting-rod is attached, for one value of a for which the port-opening is to be equalized. a and b are the positions cor- responding to the dead-points; ac is therefore equal to be or to the lap plus the lead, mn being the centre-line of the movement. Make kc ch = the lap. When the crank is in such a position that the end of the radius-rod is at /, the valve is just opening. At i the valve is just closing the same port. At f the valve is just opening to steam on the other end, and at g cut-off takes place. With g and f as centres and the length of the valve connect- ing-rod, ks of Fig. 62, as a radius, describe arcs intersecting at o, and from i and j describe arcs intersecting at /. Bisect op by the line rs. With / and q the extreme positions, draw indefi- nite arcs / and u with the same radius. Select a point on rs as a centre, such that if an arc with sp as a radius is drawn,// is equal toou. Draw sv perpen- dicular to the direction of move- ment of the valve, vsr is then the angle which the arms of the equalizing lever should make with each other. The length of the arms should be such that RADIAL GEARS HACK WORTH'S. 115 while the end of the arm rs travels from o to p, the end of the arm sv travels twice the lap. 104. Equalizing the Cut-off. The cut-off can be equal- ized by determining the points o and p from the points actually occupied by the lower end of the valve connecting-- rod when steam is admitted and cut-off at the required points instead of from the points j and i and g and /, as in the last article. While this method will make the port-opening and cut-off for one grade of expansion exactly equal, it will make them more nearly equal for all grades than if no equalizing lever is used. QUESTIONS. 131. What are radial gears? and what is the distinction made between simple and compound gears? 132. Sketch and describe the Hackworth gear. 133. How does the lead vary with different values of a? 134. Deduce the equation to the movement of the valve, and show how to draw the valve-diagram for a vertical engine. 135. What data is required and how proceed to deter- mine the other parts of the gear? 136. Explain the distinction made between right- and left- hand rotation. If the closed curve is. on your right, the open curve on your left, and the valve below the vibrating- link, is motion with the hands of the watch right- or left- handed? 137. How does the port-opening vary with the Hack- worth gear? and is the variation greater for right- or for left-hand rotation? 138. How should a horizontal engine be run with a Hackworth gear ? A vertical? Why? 139. What effect has attaching the valve connecting-rod to an extension of the vibrating-link ? 140. How can the port-opening be equalized? 141. How can the cut-off be equalized? and can it be done for several points of cut-off ? Il6 VALVE-GEARS. PROBLEMS. 50. Draw the Zeuner diagram for a Hackworth gear hav- ing r 2.5 inches, / x = 23^- inches, / 2 '= i6f inches, a = 8^ and 25. Lap = ij inches. Determine the point of cut-off. 51. Given the maximum cut-off at .7 stroke, a to be not over 24. Angle of lead 8, lap \\ inches, lead f inch, centre of shaft to centre of valve connecting-rod at right angles to stroke of piston 40 inches, the valve connecting-rod to be attached outside, find the value of / x , / , and r. 52. Given /j 2 /finches, /, = 21 inches, r= 3 T 3 7 inches, and lap = 2f inches. Required the value of a for cutting off at half-stroke, the angle of lead being 6. 53. With the data of Problem 52 and the length of the valve connecting-rod as 60 inches, determine the equalizing lever which will make the port-opening the same at the two ends of the cylinder and the cut-off at half-stroke. 54. If the crank is 16 inches and the connecting-rod 64 inches, determine an equalizing lever which will admit steam 6 before the beginning of each stroke, will cutoff at exactly half-stroke on both strokes, and will give the same port- opening on both ends, the other data being as in Problem 52. CHAPTER XIV. RADIAL GEARS MARSHALL, ANGSTROM, AND JOY. 105. Marshall's Gear. Fig. 68 is a line diagram of a Marshall gear, which is more commonly used than any other form of simple radial gear. As in Fig. 61, ab is the crank, ac the eccentric, ch the vibrating-link or radius-rod, de the valve connecting-rod, and ef is the valve-stem, i is a fixed point on the frame to which a radius-rod ig is attached, and for any one grade of expansion and direction of running ig is fixed in position, and g is therefore practically a fixed point. To g is attached a link gh = gi and to h is attached the end of the vibrating-link ch. h therefore travels in the arc a FIG. 68. of a circle jih', which replaces the straight line of the Hack- worth motion. A comparison of Figs. 68 and 61 will show that the same equation will represent the movement of the valve if we call a the angle made by the tangent of the arc jh at the point i with the centre-line ia. 106. Errors of the Zeuner Diagram. Replacing the straight line by the arc, brings another error in the diagram, 117 n8 VALVE-GEARS. which, as will be seen from Figs. 69 and 70, is quite an im portant one. Both figures show the Zeuner diagram in the FIG. 69. full lines and the actual movement of the valve in broken lines for a Marshall gear of the dimensions given in Fig. 68. Fig. 69 shows the diagrams for right-hand rotation, and FIG. 70. Fig. 70 for left-hand. In each figure A is for tan a = .75, and B is for tan a = .4. t RADIAL GEARS MARSHALL, ANGSTROM, AND JOY. 119 An examination of these figures will show that for left- hand rotation the points of cut-off and port-openings for cor- responding positions of the up and down strokes are more nearly equal. Horizontal engines should therefore be made to run in this way. For right-hand rotation the port-open- ings for corresponding positions on the up and down strokes are more unequal, and greater port-openings and more retarded cut-off are given on the up-strokes. Vertical engines should therefore be made to run in this direction, or, as is more commonly done, the eccentric could be set .with the crank, and the valve driven from ch extended. It is possible to work out a formula which gives more exactly the actual movement of the valve, but the results are too unwieldy for practical use. The chief recommendation of this gear is the small number of working parts, and the excellent distribution of steam which can be obtained by a proper proportion of the parts. 107. Proportions of Gear. The dimensions of the differ- ent parts of this gear affect seriously the points of cut-off, etc. ; and the following are given by Mr. G. A. C. Bremme, the inventor, as good proportions for the various parts : Radius-rod and arm gi (Fig. 68) = 6ac ; Eccentric-rod, ch _= 6ac ; Lead arm hd = 4.5^; Lap on steam edges (both sides equal) = .6ac. The angle a should never exceed 25. The problems give the principal dimensions of gears actually constructed which differ very materially from the proportions given above. 108. Designing. The method of designing the parts, laying down the valve-diagram, and determining an equaliz- ing lever are the same as already set forth under the Hack- worth gear. I2O VAL VE-GEARS. 109. Angstrom's Gear. Angstrom's gear is a modifica- tion of Hackworth's, and within certain limits is the exact equivalent of it. In Fig. 71 the end h of the vibrating-lever, FIG. 71. instead of being carried on a guide, is attached to a point h in a rod kl, the ends k and / swinging around the points g andy by the links gl and jk, the points^ and j being fixed FIG. 72. for any one point of cut-off. The whole arrangement forms a parallel motion. For a short distance on either side of i the motion of h is at a fixed angle tojg or to at, and when b is not allowed to travel beyond this limit the formulae and diagrams belonging to the Hackworth motion fit this gear. If, however, this limit is passed another error is introduced, f RADIAL GEARS MARSHALL, ANGSTROM, AND JOY. 121 and its amount will depend on whether the parallel motion is made as in Fig-. 71 or 72, either of which will give motion at the same angle a to ai. no. The Diagrams. If the diagrams are drawn for this gear as for the Hackworth and Marshall, it will be found that the results are much better if the parallel motion is connected as shown in Fig. 71, and right-hand rotation should be selected as giving more even cut-off and greater regularity of opening. This gear is superior to Marshall's but inferior to Hackworth's on the points mentioned, but it must be remembered that this only applies when the limits of the parallel motion are exceeded. As regards construc- tion, it is inferior to Marshall's in that it has more links, and consequently less rigidity, while it has the advantage over Hackworth's of dispensing with the sliding motion of the end of the vibrating-lever. in. Advantages and Disadvantages of Radial Gears. There is a disadvantage under which all these gears labor. To give a well-balanced motion, the vibrating-lever ch must be long ; and as all the stress, which is considerable in the case of an unbalanced valve or of high speeds, comes trans- versely on it, there is likely to be great vibration and danger of breaking. The general advantages which are characteristic of all forms of radial gears are lightness, compactness, a small number of moving parts, and constant lead. There are sev- eral forms of compound radial gears, but the discussion of one will cover the ground sufficiently. 112. The Joy Gear. Fig. 73 is a line sketch of a Joy gear as applied to a horizontal engine, ab is the crank, be the connecting-rod. To a point d in the connecting-rod is attached the link de, the end e of which swings about a point /on the engine frame, the points e and /being connected by a link ef. ig is a rod having its lower end g connected to the link de\ at its upper end / is connected to the valve-rod ij, which connects with the valve-stem at/. The point h of ig is guided along the arc of a circle Ikm, either by guides or 122 VAL VE-GEARS. by swinging around a centre. This arc is pivoted at k so that the cut-off can be changed or the engine reversed. FIG. 73. Remembering our definition of right-hand rotation, the direction shown by the arrow in Fig. 73 is left-hand. The reason for compounding a radial gear is to do away with the eccentric. The figure shows the exact motion of the points d, g and i. If the end of hg were attached directly to the connecting- rod at d, the point d during the lower half-revolution of the crank would describe the lower half of a curve which nearly coincides with a circle having k as a centre, so that h would have little motion along Im, and the only motion given to the valve would be that due to the oscillation of hi about k. By causing g to describe the irregular oval, the travel of the RADIAL GEARS MARSHALL, ANGSTROM, AND JOY. 12$ point h above and below the centre of suspension k of the arc Im is equal and approximately symmetrical. 113. Movement of the Valve. When the engine is on either dead-point, call the angle the link ed makes with a vertical line, and the angle kg makes with the vertical. Call the distance de = /, dg = / 4 , hg =. l lt ki = / 2 . Then / 4 vers = /! vers 0, as the angularity of the rods should neutralize each other in a well-designed gear. Also, / 4 sin + ^1 s i n ^ radius of crank, as the point ^ should travel symmetrically under k. Also /, sin 6 R. In designing, the most convenient assumption is that which fixes the value of / 4 and I, . The distance from the centre of the valve-stem to the axis of the cylinder is fixed, and is approximately / 2 + A / 4 . Having assumed these values, we have / 4 (i cos B) = /,(i cos 0), / 4 sin 6 -f- /, sin = R and / 3 sin # = R. From these equations, by eliminating 6 and 0, we can get a value of / 3 in terms of / 4 , / x , and 7?. In the Hackworth gear the movement of the valve is composed of two parts, one part mq, in Fig. 62, due to the movement of i along ice, and one part qk due to the turning of ir around i. In the Joy gear, Fig. 73, the point d moves vertically a distance -rR sin GO when c = the distance cd and b = cb, and d is a distance R cos &? from its middle position. g has moved vertically practically the same distance as d, or -R sin GO, and horizontally is 3 . -R cos 00 from its middle position. If, then, instead of r sin GO in the equation to the Hack- worth motion, we put -=-R sin GO, and instead of r cos GO we put 3 -7? cos CB?, we get the equation to the movement of the valve. 124 VALVE-GEARS. As i is beyond h, the equation for the Hackworth mo- tion is x = j r cos oo-\-r (^, - tan a J sin GO and making the substitutions noted above, we have X = IL= ^-RJ cos oo + R (^^ tan ) j sin ,. from the movement of the valve from its middle position. The co-ordinates of the centre of the valve-circles are hori- zontally l -*^Rj> and vertically ^^(^y- /a tan a}. From these values the valve-circle can be drawn, and the points of cut-off, steam admission, etc., determined. 114. Errors of the Zeuner Diagram. The errors in using the diagram are shown in Figs. 74 and 75, where the FIG. 74- arrows indicate the movement of the crank : that is, Fig. 74 is for left-hand rotation by definition, and Fig. 75 is for right- hand rotation, A being drawn for tan a = .75, and B for tan a .4. It will be seen that the lead is equal on both ^nds of the cylinder for motion in either direction, but the cut-off is more nearly equal for right-hand rotation than for left. RADIAL GEARS MARSHALL, ANGSTROM, AND JOY. 12$ QUESTIONS. 142. Sketch and describe a Marshall gear. 143. Deduce the equation for the movement of the valve with the Marshall gear. 144. What parts of the valve movement are most affected by the errors of the Zeuner diagram ? 145. How should a horizontal engine run with a Marshall gear? A vertical? Why? 146. Explain the method of designing a Marshall gear. FIG. 75. 147. Sketch Angstrom's gear, and what is the objection to it? 148. What are generally the advantages and disadvan- tages of radial gears? 149. Sketch a Joy gear. 150. Determine the movement of the valve in the Joy gear. 151. Is the Zeuner diagram a close approximation in the case of a Joy gear? 152. If the smaller opening of the port is desired on the end opposite that which the gear as connected would give it, what must be done to change it? 1 26 VAL VE-GEARS. PROBLEMS. Problems 50, 51, and 52 apply equally well to the Mar- shall gear. 55. Solve Problem 53 for a Marshall gear, having the reversing-arm and radius-rod each 18^ inches. 56. Solve Problem 54 for a Marshall gear having the re- versing-arm i8 inches. 57. In a Joy gear having crank 8 inches, =13 inches, b = 39 inches, /, 16 inches, l z = 8 inches, / 3 = 24 inches, and / 4 8 inches, the lap being 2 inches, and the reversing- arm and radius-rod 12 inches, find the value of a for cutting off at GO 90, the angle of lead being 8. 58. Draw a diagram for the Joy gear in Problem 57, showing the error of the Zeuner diagram for cutting off at oo ~ 90 on both strokes. 59. If the engine is to cut-off exactly at half-stroke on both ends, what must be the lap and the lead on each end. 60. In a Marshall gear having r = 5J- inches, lap i", /j == 66, and / 2 = 30, the reverse lever 30 and tan a = .6. Find the points of cut-off in both ends of the cylinder from the Zeuner diagram, and by laying down the gear, if the stroke is 48 inches; and the connecting-rod 100 inches. CHAPTER XV. DOUBLE VALVES GRIDIRON VALVE. 115. Kinds of Double Valves. We have already seen that in the case of a single valve moved by an eccentric, early cut-off makes equally early admission, and affects the exhaust in the same way as the steam. To retain the proper opening and closing of the exhaust while varying the cut-off, double valves are used, and may be divided into two general classes. In the first class the second valve called the cut-off valve moves on an independent valve-seat, and controls the admission of steam to the steam-chest of the main valve. By cutting off the admission of steam to the main valve- chest, steam is prevented from entering the cylinder whether the main valve is open for steam or not. As the exhaust takes place under the main valve, no change is made in the exhaust. In the second class the cut-off valve moves on the back of the main valve, and produces the same effect. 116. Gridiron Valve. The first class is represented in Fig. 76, which is a sketch of a Gonzenbach or gridiron valve, a is the main valve, b is the valve-chest, c is the seat of the cut-off valve, and d the cut-off valve itself. Steam first enters the space e above the cut-off valve, and when this valve is open passes through the ports g and h, into the main valve chamber b. If the valve a opens, steam then passes into the cylinder. After steam is cut off by the valve d, the steam in the chest b and in the cylinder expands as one volume; and after the main valve closes, the steam in the cylinder alone expands. Exhaust takes place under the valve 127 128 VALVE-GEARS. a as in an ordinary slide. The valves a and d are moved by independent eccentrics, but the valves never occupy the FIG. 76. relative positions shown when connected to their eccentrics. The small diagram shows the relative position of the crank and the two eccentrics. 117. Polonceau Valve. A valve of the second class is shown in Fig. 77, which is a sketch of Polonceau's valve. The valves are in their middle position at the same time a FIG. 77. position they never assume if connected to their eccentrics, a is the main valve, which has two passage-ways b and c, the DOUBLE VALVES GRIDIRON VALVE. 129 part between the passage-ways being an ordinary D-slide, On the back or tops of the main valve is a cut-off valve d r which is a plain flat block. Each valve is moved by its own eccentric. As the main valve moves to the left steam passes from the chest e through the passage 3, to the steam-passage lead- ing to the right-hand end of the cylinder. The block d is- moving at the same time, and by a proper setting of the eccentrics, at the right point in the stroke the valve d covers the passage b, and steam is cut off. The exhaust here as in the other class takes place independently of the admission and cut-off. The small diagram shows the relative position of the crank and the two eccentrics. 118. Diagram for Gridiron Valve. To determine the point of cut-off, etc., of the Gonzenbach valve, we proceed FIG. 78. as follows : In Fig. 78 draw the diagram for the main valve as for any single valve. As the exhaust is not affected by the cut-off arrangement, we will omit the exhaust lines from the diagram. In the figure A is the admission and B the J30 VAL VE-GEARS. cut-off in tne right-hand end of the cylinder, C is the admis- sion, and D the cut-off for the left-hand end. Suppose the cut-off valve to be moved by an eccentric without angular advance. Fig. 79 will then represent the dis- tance the cut-off valve has moved from its central position for any position of the crank. If the width of the port h is e and of g is /, it is evident that if the valve moves away from its central position a distance e+f = s, the edge of d is just closing the port h. If in Fig. 79 we draw a circle efg with s as a radius, the cut-off valve covers the port while the crank is passing from E to F. Referring to Fig. 78, it is evident that the cut-off valve must open between D and A in order that the steam may be admitted as soon as the main valve opens. It must close between A and B to cut off before the main valve does, and it must not open again until the crank passes B, otherwise steam would be admitted twice during one stroke. The cut-off valve must again open be- tween B and C to admit steam to be ready for the next stroke. That is, in a combined diagram for the two valves E should fall between A and B, and F should fall between B and C, and this generally requires that the angle of advance of the cut-off valve should be negative. 119. Combined Diagram for both Eccentrics. Fig. 80 represents such a combined diagram. Steam is admitted through the cut-off valve at F and the main valve opens at A. Cut-off by the cut-off valve takes place at E and by the main valve at B. The cut-off valve opens again at F, and FIG. 79. DOUBLE VALVES GRIDIRON VALVE. 131 the main valve to the other end of the cylinder at C. As the point F must always fall between B and C y the limits of cut-off obtainable by this gear are not very great. If E falls on R, the main and cut-off valve close at the same time ; and if F falls on C, the cut-off takes place at lt FIG. 80. and the cut-off can therefore be changed by changing the value of s between oe' and oB. 120. Limits of Cut-off. In the diagram we have drawn, the centre of the cut-off valve-circle falls on oB. If it falls on any other line, as oB', the limits of the cut-off would be less. For if ol' is such a value of s that it cuts the cut-off valve-circle on oB, the cut-off takes place on oE^ , and any greater value of s will cause the valve to admit steam twice 132 VALVE-GEARS. during the stroke. The range of cut-off is then from oE z to oE^ . This, while allowing an earlier cut-off, gives less range, and at the same time makes a bad distribution of steam if a later cut-off than oE^ is required. It is generally better to draw the diagram so that the centre of the valve-circle for the cut-off valve is on the line of cut-off for the main valve. 121. Width of Ports. In laying down the diagram all the data of the main valve are given, or can be determined as already shown for a plain slide. The area of the port through the seat of the cut-off valve should be as great as and preferably a little greater than the area through the main valve-seat. This will determine the value of e. The opening in the cut-off valve is usually greater than this. The reason for this is, that the valve may be opened and closed quickly, or for the same reason that the over- travel is given to an ordinary slide-valve. Evidently the port begins to close when the valve has travelled a distance -Ll - from its middle position, is entirely closed when x s, begins to open again when x is equal to s, and is wide open, again when x = . 122. Angle of Advance. Having determined e and /, suppose in Fig. 81 we have the main valve-circle given, cutting off at ob, and suppose it is desired with the cut-off valve to cut off steam when tne crank is at oe. With a radius equal to s, draw the circle fgh. Then, in order that the cut-off valve will not admit steam again before the main valve cuts off, the valve-circle for the cut-off valve must pass through /, r x (s e) or <^> r x -\-e s. The distance between the steam-passages on the top of the main valve = d-\- 2(s e). The outer wall of the steam-passage b should be so wide that for the greatest relative travel the right-hand edge of d should pass over the end of the valve or be < r x s. 142 VA L VE- GEA RS. QUESTIONS. 163. What is meant by the relative valve-circle, and how is it determined ? 164. Prove that the relative movement of the two valves is given by the relative valve-circle. 165. Describe the Polonceau valve. 1 66. What arrangement was used for moving the Polon- ceau valves? 167. How is the valve-diagram for a Polonceau valve and gear drawn ? 1 68. What are the limits of cut-off with this gear? 169. How are the dimensions of the main and cut-off valves determined? PROBLEMS. 64. A Polonceau valve is driven by a Gooch motion having r 2| inches, g 57 inches, d 27, and lead equal to \ inch, and for u = 6 the cut-off just opens as the main valve closes. Length of link 12 inches. Draw a diagram showing the cut-off for u = 6 inches, #, = 2, 4, and 6 inches. 65. Determine the greatest and least limit of the cut-off with the data of Problem 64. 66. If the port in the main valve-seat is if inches and in the cut-off valve-seat is if inches, with the value of s as found for Problem 64 draw the valve in the position it would occupy if GO 40 and the cut-off at stroke ; the exhaust- port in the main valve-seat being 4 inches and the bridges i j- inches, the iron being -J inch thick in the valves. CHAPTER XVII. BUCKEYE GEAR. 135. The Valve. The Buckeye Automatic Engine valve resembles the valve we have been dealing with, but its method of action is entirely different. Fig. 86 is a part sec- tion through the valves and cylinder, a is the main valve, having ports , through which steam passes from the chest h to the passage c in the cylinder, e and e are two blocks k\\\\> FIG. 86. connected by rods /passing through an opening in the main valve, and forming together the cut-off valve. As steam is admitted through //, and the exhaust takes place at g, the action of the main valve is the same as of an ordinary D- slide taking steam inside and exhausting outside. 136. The Eccentrics and Connections. In Fig. 87 if ab is the crank turning as shown by the arrow, an eccentric ac FIG. 87. connected directly, by an eccentric-rod cd, to a valve-stem de would give the mam valve the proper motion. At the point d, the eccentric-rod is also connected to the 143 1 44 VAL VE - lever df, which is pivoted to the frame of the engine at f t i is the middle of df, and to i is attached the centre of a vibrating-lever hj, one end of which, /, is connected to the cut-off valve-stem /, and the other end, h, is connected by the eccentric-rod hg to the cut-off eccentric ag. Assume that ac and ag are in their middle position when at ac' and ag' . Call c'ac = tf and g'ag = d, . Call ac = r and ag = r l (in the Buckeye engine these are equal). Evidently, if both eccentrics are in their middle positions at the same time, the valves would occupy the positions shown in Fig. 86; but as the eccentrics are never in their mid-positions at the same time, the valves and ports are never, in the engine, as shown in the figure. 137. Movement of the Valves. When the crank ab has moved in the direction of the arrow through the angle co, if the eccentrics are in their proper position the angle cac = d -\- GO and g'ag = ^ GO. The main valve has moved to the left a distance r sin (GO-}- d). The point d has moved to the left the same distance, and the point i has moved to the left sin (GO -f- 6). If the eccentric ag is still in its mid- position the lever hj turns about h as a centre, and j has moved to the left a distance = r sin (co-\- #), and the cut-off valve has moved to the left a distance r sin (GO + d) due to the movement of the main eccentric alone. To bring the cut-off eccentric to its proper position g moves to the right a distance r, sin (GO tfj and /moves to the left a distance r t sin (GO #,), and the total movement of the cut-off valve is r sin (GO-}- 6) -J- r l sin (GO 146 VAL VE-GEARS. and m. This will be found to be at or before the beginning of the stroke, and of course no such movement of the eccen- tric is necessary. If the angle of advance ^ is zero, the cut-off takes place at on, and the port is again open to steam at of after the main valve has cut off at ok. As d l increases negatively the point of cut-off is later and later, and if necessary could be made as late as with the main valve by allowing the cut-off valve to change its angular advance sufficiently. 140. The Governor. The arrangement by which the change in angular advance is effected is shown in Fig. 90. FIG. 90. A. governor-wheel is keyed to the shaft ; a and a are two arms pivoted to the arms of the wheel at gg ; bb are rods BUCKEYE GEAR. 147 joining these arms to the lugs cc, which are attached directly to the cut-off eccentric. This eccentric is loose on the shaft while the eccentric for the main valve is secured at its proper angle of advance; dd are weights which, as the speed increases, fly out and turn the cut-off eccentric, thus changing the angle of advance; e and /are springs which act against the centrifugal force tending to throw the weights outward. QUESTIONS. 170. Make a sketch of the Buckeye valve, and show the relative position of crank and eccentrics which would drive the valve as a Polonceau is driven. 171. Make a sketch of the eccentric and connections when the engine is intended to run in the direction opposite to that shown in Fig. 87. 172. Deduce the equation to the relative moments of the two valves. 173. Draw the cut-off valve-diagram, and explain how the point of cut-off is changed. 174. Sketch and explain the governor. PROBLEMS. 67. In a Buckeye gear if the lap is i^ inches, the port- opening is if inches, and the maximum cut-off is to take place at \ stroke, what must be the travel of the main valve ? If the cut-off eccentric has the same eccentricity, what must be the angular movement of the cut-off eccentric to vary the cut-off from \ to f the stroke, and what must be the actual value of # for cutting off at f stroke, s being I inch ? 68. If the value of s is changed, does it make any differ- ence in the angular movement of the eccentric? Can the value of s be made too large or too small ? How then would you choose the proper value ? What should it be in the last problem ? 69. What must be the width of the cut-off blocks in the last problem ? CHAPTER XVIII. MEYER VALVE AND GUINOTTE GEAR. 141. The Meyer Valve. We have seen that by chang- ing the position of the cut-off eccentric the point of cut-off can be changed. There is another device by which this can be done, which is shown in the Meyer valve, represented in Fig. 91. a is the main valve, having passages through it like the Polonceau valve, b and c are the cut-off blocks, FIG. 91. which together form the cut-off valve. The valves are moved by independent eccentrics connected directly. The valve-diagram for this valve is the same as for the Polonceau. b and c are connected by a right- and left- handed screw by means of which the distance between the blocks is regulated. 142. Changing the Distance between the Blocks. To determine what effect the changing of the distance between 148 MEYER VALVE AND GUINOTTE GEAR. 149 the blocks has on the point of cut-off, let Fig. 92 be the valve- diagram, supposing the main-valve eccentric to be set with an angle of advance abe, and the cut-off eccentric with an angle of advance abd. Then, as already shown, for the Polonceau gear the relative, valve-circle is that shown at FIG. 92. cfghb. The cut-off takes place when the distance be, Fig. 92, is equal to s of Fig. 91. Suppose now in Fig. 91 that the blocks b and c are moved closer together. The distance s in Fig. 91 is increased, and in Fig. 92 if bf the new value of s, the cut-off now takes place at bj. As in a gear of this kind it is convenient to measure the distance between the blocks, let it be 2y. Let the length of b or c be d, and the distance between the out- side edges of the ports be L. Then 2s L 2d 2y. If it is required to know at what point the blocks should be for any point of cut-off : 2y = L 2d 2s, or y = d - s. Drawing a circle ijh', Fig. 92, with d as a radius, at any point of cut-off, as be, we have be = d = y -f- s, and be =^ s, ce = y. As the cut-off becomes earlier and earlier ISO VAL VE-GEARS. at bl, we have s o and y = bl\ and as the cut-off is still earlier as bm, s is negative and bn and y = ;#;/. So that if the value of y is changed, almost any point of cut-off desired can be obtained. It is to be remembered that the pointy or //, in which the circle with s as a radius cuts the relative valve-circle a second time, must not occur before the main valve cuts off, otherwise steam is admitted twice during the same stroke. 143. Designing a Meyer Valve. In designing a gear of this kind, the main valve is designed for the latest cut-off desired, and the range over which the cut-off valve is in- tended to act is determined. Suppose in Fig. 93 cab to be FIG. 93. the angle of advance of the main valve, ab its half-travel, and ae the lap. Steam is admitted at af, and cut off by the main valve takes place at ag. Now suppose it is required to design a Meyer gear to act between ah and ag. As the cut- off is to range entirely up to that of the main valve, and as we do not want any larger eccentric than is necessary, we will assume that the relative valve-circle centre falls on ag. We will assume further that the engine is intended to run in either direction equally well. That this may be so, the centre of the valve-circle for the cut-off valve should be on at. Through b draw bi parallel to ag. Then at is the eccen- tricity of the cut-off valve-eccentric, and bi is the equivalent of the diameter of the relative valve-circle. We have seen that when s is greatest the cut-off is latest, or s must be t MEYER VALVE AND GUINOTTE GEAR. 151 greatest at ag. The blocks can here be close together, or y = o, and we have - d = s = bi. When s is least the cut- off is earliest ; and we have from the diagram the value of L s = ak = d y, or y 4- ak -\- ^- d\ = ak + bi = ak + ah = kh. From Fig. 91 , = y -\- d -{- s, or d -\- s = - y. 2 2 144. Length of Cut-off Blocks. The relative valve-mo- tion should never be so great that the left-hand edge of c uncovers the left-hand edge of the passage in the right-hand end of the main valve. Calling the port at the top of this passage e, we have d-\- s r x > *, or - y r x > e, or -> Now the greatest value of y is kh, so that should be greater than e -j- kh -f- ah. This determines the value of , and if the blocks are to come together for greatest cut-off. The value of y depends on the desired point of cut-off, and is determined as already shown. 145. Cut-off with Inside Edges. We have made the valve cut off with the outside edges. By setting the eccen- trics properly, the cut-off could have been effected by the inside edges; but any such arrangement is of no advantage, as the main valve and chest are increased in length, and no better distribution of steam is effected. 152 VALVE-GEARS. 146. Guinotte Gear. There are numerous other ways in J which the variation of cut- off can be effected with a Polonceau valve or a modi- fication of it, but we will only take up one more. Referring to Fig. 85, as a Gooch link is there used, the centres of the valve- circles move in a line at right angles to os, and the centres of the relative valve- circles fall on the line om. Now if by any means the centre of the cut-off valve- circle can be made to move at any other angle than a right angle to os, the centre < of the relative valve-circle will no longer always be on om, and a greater variation of cut-off can be obtained. There are two methods by which this can be done, one of which is given here. Fig. 94 represents a Gooch motion similar to Fig. 46, except that the eccentrics are set with unequal angles of advance, and the unequal eccentrics, lengths of eccen- tric-rods, and unequal parts into which the link 'is di- vided at the point of sus- pension, replace the cor- responding equal ones in Fig. 46. f MEYER VALVE AND GUINOTTE GEAR. 153 147. Movement of the Valve. We have, as in that case (calling everything- pertaining to the upper end of the link sub i ( , ) and to the lower end sub 2 (^ ) g moves to the right because of the movement of the upper eccentric, and because of the movement of the lower eccentric, and the total movement of g is therefore the sum of these two, or * = r > sin < M + 6 > + *> + r < sm <* - * + *> This can be expanded as in the case of the Gooch motion, and becomes 7 L ZT r ( sin * + 7" cos *) f 6 i ~r 6 2 \ s* i ) cos r 2 fcos d 2 sin (^,j > sin co; and we have for the valve-diagram x A cos GO + B sin G?, in which ^4 and B are given from the preceding equation. The first power of u appears in both of these values, and therefore the centre of the valve-circles moves in a straight line inclined to os. 154 VALVE-GEARS. 148. To draw the Valve-diagram. To draw the line of centres a convenient method is as follows: First suppose u c ; then and which gives the centre of the valve-circle for an ordinary Gooch motion having data r 1 , d l , c l , and g l when u = c l . Now let u = a , and we have and which gives the centre of the valve-circle for an ordinary Gooch motion, with data r^ , d^ , 2 and^ when u = c^ . The line can then be drawn through these points and the points of cut-off for different values of u for the cut-off radius-rod found, the main valve-eccentric having data r and d. Fig. 95 shows the construction of the diagram for the main valve and for the cut-off valve when u = c t , - 1 , o, and 2 . As this form of gear is only of use for engines run- ning in one direction, it is not at all likely to be used to any extent. QUESTIONS. 175. Sketch a Meyer valve, and explain the method of drawing the valve-diagram. 176. Explain from the diagram the effect of changing the distance between the cut-off blocks. A, = r, (sin d, + ^ cos ,, \ o a ^ r^ [cos t sin Bilgram diagram, ......... 149 157 " Problems solved by the, ..... 150 158- Bridges, Thickness of, - 35 4<> Buckeye engine eccentrics and connections, ... 136 143 " " Movement of valves in, . . . 137 144 ' " Varying cut-off in, 139 145 " governor, ........ 14 I 4^ " valve 135 143 171 3/2 INDEX. C ART. PAGE Centre of valve travel, To lay down the, *.-. 59 67 To put engine on the, ....... 38 45 Changing angular advance for regulating, . . , . 85 98 " eccentricity for regulating, ...... 86 99 Circular diagram f ^r piston position, ...... 34 37 Comprersi n and exhaust, Equalizing, 33 36 Movement of piston during, . . . . 17 15 P-riod of, 17 I4 Crossed rods, 55 61 Curve of centres for Stephenson link, ...... 47 56 Cut-off, Arrangement for varying, with gridiron valve, . . 124 135 " blocks of Meyer valve, 142 148 " " " " Length of, 144 151 " Equalizing, 31 34 " Equalizing, with radial gears, ...... 104 115 " Stephenson's link 57 65 " in Buckeye engine. Varying, ...... 139 145 '* Limits of, with gridiron valves, . . . . 120 131 " " " Polonceau gear, 133 140 " Point of, 12 n " valve diagram for Buckeye gear, . . . . .138 145 " Width of gridiron, ... 125 135 " with gridiron valve, Varying, ...... 123 133 D Designing a double-ported valve, 26 28 Fink motion, ........ 82 94 Gooch motion, ....... 71 82 " Hackworth gear, .......97 no " Marshall gear, 108 119 Meyer valve, ........ 143 150 " plain slide-valve, 35 40 " " " approximate solution, . . .36 41 " " " with equalizing lever . . 37 43 " Stephenson motion, 50 58 Distribution of steam from the diagram, . . . . . 16 14 Double valve, . . 115 127 ported valves, 26 28 Eccentricity, Average valves of, ....... 35 40 Effect of changing, ....... 25 25 " Regulating by changing, 86 99 INDEX. 173 ART. PAGE Eccentric rod, Length of, 51 58 " Setting the, 39 45 ' The, 3 2 " " virtual, 49 58 " To determine the position of the, 24 24 Eccentrics and connections of Buckeye engine, . . , . 136 143 Elliptical diagram, 153 163 Equalizing bell-crank leyer, ....... 32 35 " cut-off, . . 31 34 " " and lead, 32 35 " with radial gears, 104 115 " " Stephenson's link, 57 65 " exhaust and compression, ...... 33 36 " lead with Stephenson's link, ..... 56 63 " port opening with radial gears, 103 113 Erie engine governor, 88 loo Error of the Zeuner diagram for the Joy gear, . . . .114 124 " Marshall gear, . . .106 117 " " " " " Stephenson gear, 63 70 Exhaust and compression, Equalizing, 33 36 Exhaust lap, 5 4 " lead, 14 12 " Movement of piston during, . . . . . 17 15 " passages, I i " Period of, 17 14 " ports, Area of, 35 40 " Width of, 35 40 Expansion, Movement of piston during, 17 15 " Period of, 17 14 F Fink motion, . 74 87 u Designing a, . . 82 94 " Hanger for radius- rod with a, 80 92 ' " Lead with a, 75 87 * " movement of valve with a, Equation for, . . 77 89 *' " Radius of link of a, 75 87 " " Setting the eccentric of a, ..... 81 92 " Suspension of link in a, ...... 76 89 " " Valve diagram for a, ...... 78 91 Four valves, 30 31 G Gonzenbach valve (see Gridiron), 116 127 Gooch link-motion 64 75 INDEX. ART. PAGE Gooch Link-motion applied to Polonceau valve, .... 131 139 " " Designing a, 71 82 " " Lead with a, . 66 78 " " Movement of valve, equation for, 65 76 " " Radius of link in a, ..... 67 78 " " Suspension-rod for a, 68 79 " " The hanger for a, ..... 69 80 " " Valve diagrams for a, . . ... .70 80 " " with unsym metrical parts, .... 146 152 Governing by changing angular advance, 85 98 " " eccentricity, 86 99 " " " and angular advance, . .87 99 Governor, Armington and Sims, 89 101 " Ball engine, . .... ... 90 104 " Buckeye engine, c 140 146 " Erie engine, .88 100 Governors, Throttling, 84 98 Gridiron valves, . 116 127 " " Angle of advance with, 122 132 " " Arrangement for varying cut-off with, . . 124 135 ' " Combined diagrams for, . . . . 119 130 " " Diagrams for, 118 129 " " Limits of cut-off with, ..... 120 131 " " Varying cut-off with, ...... 123 133 " " " width of cut-off valve, .... 126 135 " " width of cut-off valve, ... .125 135 " " Width of ports with, ...... 121 132 Guinotte gear, 146 152 " " Movement of valve with, 147 153 " " Valve diagram for, ..., 148 154 H Hackworth gear, 93 107 " " attaching valve-stem outside, . . , . 102 113 " " Connecting vp a, . . 101 113 " Designing a, ..... , 97 no *' " Equation to movement of valve with, . . 95 108 " " Error in Zeuner diagram for, . . . . 99 in " " Valve diagram for, . . ... 96 109 " " Variation in port opening with, ... 100 113 Hanger for radius-rod, Fink motion, . . . . 80 92 " " Gooch motion, 69 80 Stephenson's link, . 53 59 INDEX. 175 ART. PAGE Joy gear, The, 112 121 " " Equation to movement of valve with, . . , . 113 123 L Lap, 5 4 " Average values of, 35 4 " different on the two ends of valve, . . . 31 34 " Effect of changing, ,..25 25 Lead, 14 12 " and cut-off, Equalizing, ....... 32 35 ' ' Average values of, ........ 35 40 " Equalizing, with Stephenson's link, 56 63 " with Fink motion, 75 87 " " Gooch motion, ....... 66 78 " " radial gears, 94 108 Length of eccentric rod, ....51 58 " " link, .........* 52 59 " " valve stem, 51 58 Link block, Slip of, 42 48 " Length of, 52 59 " motion, Allen, 72 85 Fink, 74 87 " Gooch, 64 75 " " Reducing slip in a, ...... 62 70 * f " Stephenson's, 40 47 " To determine centre of suspension of hanger, . 60 68 " " To lay down a, 58 67 " " To lay down centre of travel of valve, . . 59 67 Link motions, , 40 47 " Radius of Fink, 75 87 " " Gooch, ........ 67 78 " Stephenson's, ....... 43 50 " Stephenson's, Point of suspension of, 41 48 Links, Kinds of . 44 50 M Marshall gear, 105 117 " Designing a, 108 119 " Equation to movement of valve with a, . . 105 117 " " Proportions of, ....... 107 119 Meyer valve, 141 148 " " Changing distance between block of a, ... 142 148 " " Cutting off with inside edges, . . . . . I4S 15* 1/6 INDEX. ART. PAGE Meyer valve Designing a, 143 150 " " Length of cut-off blocks of, ..... 144 151 Movement of upper end of hanger, Stephenson's link, . 53 59 Movement of valve, Equation for the, with Buckeye gear, . . 137 144 " " " Fink gear, . . 77 89 " " " Gooch gear, . . 65 76 " " " " Hackworth gear, . 95 108 " Joy gear, . . .113 123 " " " " " Marshall gear, . 105 117 " " ** " " single eccentric, 9 7 " " " Stephenson gear, 45 52- O Opening of port, 12 10 Open rods, 55 61 Over-travel, 18 19 P Piston movement, Circular diagram for, . . 34 37 Piston valves, ........ 28 30 Point of suspension of link, ...,.,. 41 48 Polonceau gear, 131 139 " " Limits of cut-off with, 133 140- " " Valve diagram for 132 139 " valve, ...... o .. 117 128- 130 W " " Dimensions of, .... 134 141 Porter-Allen engine, regulating a, 83 9$ motion, ......... 83 94. " " radius of link, . 83 95 " valves = 30 31 Port opening, Equalizing, with radial gears, .... 103 113 " '* variable in Hackworth gear, ..... 100 113 Ports, area of, .......... 35 40 " Width of, with gridiron valve, ...... 121 132 Position of stud on Stephenson's link, 6l 68 Put engine on centre, to, ........ 38 45 R Radial gears, ... Q2 107 " " advantages and disadvantages of, . . . . in 121 " Angstrom's, .....*... 109 120 " . Definition of right-hand rotation with, . . .98 ni " Hackworth's, 93 107 INDEX. I? 1 ? ART. PAGE Radial gears, Lead of valve with, 94 io& " " Joy's, 112 121 " " Marshall's, 105 117 Radius of link, Changing, for Stephenson's gear, ... 56 65 " " Fink motion, 75 87 " " Gooch motion, ....... 67 78 " " Porter-Allen motion, 83 95 " " Stephenson motion, 43 50 Reducing slip, 62 70 Relative movement of two valves, 127 137 " valve circle, 128 138. " " " To draw the, 129 138 Rculeaux diagram, 151 160 " " Solution of problems by the, . . . .152 161 Reverse lever, effect on movement, ...... 23 24. " " To design equalizing, ...... 37 43. S Setting the eccentric, 39 45 " " " with a Fink motion, 81 92 Slip of the link-block, 42 48 " Reducing 62 70 Steam inside the valvj, Taking, 29 30 " lap, 5 4. " passages r i " ports, I i " Area of, . 35 40 Stephenson link-motion, 40 47 " " Changing radius of link, ... 56 65 " " Curve of centres, ..... 47 56 " " Designing a, 50 58 " Equalizing cut-off with a, ... 57 65 " " Equalizing lead with a, . . . .56 63 ** ** Equation to movement of valve with, . 45 52- " " Error of Zeuner diagram for, 63 70 " " Hanger for link of, 53 59, " " To lay down a, 58 67 '* " To draw valve diagram for, . . .48 57 " " Valve diagram for, .... 46 55 Stephenson link, Point of suspension of, ..... 41 48 " Position of stud on a, 61 68 " " Radius of, 43 50 " " suspended at bottom, Hanger for, ... 54 60 " " " centre of chord, Hanger for, . 54 60 " " with crossed rods, 55 6l Straight link, . 72 85 178 INDEX. PAGE Throttling governors, . . . 84 98 Trick valve, ,.27 28 Two valves, 30 31 V Valve, Allen or Trick, ... 27 28 Armington and Sims, 28 30 " Balanced slide, .... 27 28 " Ball engine 91 105 " circle, Relative, 128 138 " " To draw the relative, . 129 138 " D slide i i '" " method of action of, ...... 2 2 " " movement of, ...... . . 7 5 " Designing a plain slide 35 40 " " " approximate solution, ... 36 41 " " " with equalizing lever, ... 37 43 *' diagrams, 9 6 " " Bilgram, ........ 149 157 *' " Buckeye engine, 138 145 Elliptical, i3 163 '* " Error in, for Hackworth gear, . . . 99 m " " " " Stephenscn gear, . . . 63 70 " " " Joy gear, 114 124 " " " Marshall gear, . . . . 106 117 " " for a given engine, To draw, .... 15 13 " " " Allen motion, 73 85 " ' " Angstrom gear, HO 121 '* " " each end of the cylinder, . . . , 17 14 " " Fink motion 78 91 * ' " Gooch motion, 70 80 ' " gridiron valves, Il8 129 * " " " Combined, . . . .119 130 te " Guinotte gear, 148 154 '* " " Hackworth gear, .... * 96 109 " '"' " Polonceau gear, 132 139 " " " Stephenson gear, ..... 46 55 " " " " To draw, ... 48 56 " " Reuleaux, . 151 160 " when 5 = o, 10 7 " Zeuner, 9 6 * " ir 10 *' Dimensions of Polonceau, 134 141 *' Double 115 127 INDEX. 179 ART. PAGE Valve, Double ported, 26 28 Equation for movement of a 9 7 " " with angular advance, . .13 n " face, 4 3 " " not parallel with piston travel, .... 23 23 " for Ball engine, 91 105 " for Buckeye engine, . . . . . . 135 143 " Gridiron or Gonzenbach, 116 127 " gear, Guinotte, ........ 146 152 " " Polonceau, 131 139 " Meyer, 141 148 " Piston, 28 30 " Plain slide, i i " Polonceau, .117 128 130 139 " Porter-Allen, .... c .... 30 31 " Position of, for any position ot piston, .... 6 4 *' seat, 4 3 " Setting the, . 39 45 " stem, Length of, 51 58 " To adjust the, 39 45 " Velocity of movement of, 154 t6 4 Virtual eccentric, The, . . .... 49 58 Yoke connection, SHORT-TITLE CATALOGUE OF THE PUBLICATIONS JOHN WILEY & SONS, INC. NEW YORK London: CHAPMAN & HALL, Limited Montreal, Can.: RENOUF PUB. CO. ARRANGED UNDER SUBJECTS Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only. All books are bound in cloth unless otherwise stated. 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